<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K8" target="_self">(#)</a> <font color="Maroon">seq2</font> implies for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K8" target="_self">(#)</a> <font color="Maroon">seq2</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E2:25_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<i><font color="Green">E30</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font> = 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K9">/\</a> <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">seq2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">seq1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">seq2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E4:25"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font> holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:25"/>
<font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K8" target="_self">(#)</a> <font color="Maroon">seq2</font>
 <b>by </b><i>, </i>;<br/>


</div>
